Your operation could maintain 2 sets of cash flow: one for patient care, and one for administration. Maybe you could qualify for NGO status and get some government to fund some of the administration side.
This would be somewhat similar to Costco's model, where most (all?) of employee wages come from membership card fees, not product revenue. With this model, you could still let people donate 100% of their funds to the patient if they wanted.
You could even make a slider with an animation that displays the percent which goes to each part of your business - so 10% or 15% of the donation could goto administration.
I also strongly suggest this humblebundle ui on 'Question 2' on the bottom. This would be ideal... HOWEVER, there also is the possibility you end up having not enough for the patients and plenty for the salaries.
Maybe make it 'what is the max percent you want to see spent on admin cost'. that way you are allowed to use as much as you'd like to for the patient.
Those are awesome suggestions. Thank you! We were actually just approved as a 501(c)(3) non-profit organization in the states, and that will open us up to a lot of funding opportunities. But we love the Costco Model idea!